Theorem elrnmpt2d 5958

Description: Elementhood in the range of a function in maps-to notation, deduction form. (Contributed by Rohan Ridenour, 3-Aug-2023.)

Hypotheses

Ref	Expression
elrnmpt2d.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
elrnmpt2d.2	⊢ (𝜑 → 𝐶 ∈ ran 𝐹)

Assertion

Ref	Expression
elrnmpt2d	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐴(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem elrnmpt2d

Step	Hyp	Ref	Expression
1		elrnmpt2d.2	. 2 ⊢ (𝜑 → 𝐶 ∈ ran 𝐹)
2		elrnmpt2d.1	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3	2	elrnmpt 5950	. . 3 ⊢ (𝐶 ∈ ran 𝐹 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥 ∈ 𝐴 𝐶 = 𝐵))
4	3	ibi 270	. 2 ⊢ (𝐶 ∈ ran 𝐹 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)
5	1, 4	syl 18	1 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 𝐶 = 𝐵)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2146  ∃wrex 3091   ↦ cmpt 5194  ran crn 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  df-cnv 5671  df-dm 5673  df-rn 5674

This theorem is used by:  ablsimpg1gend  20225  elrgspnlem4  33633  elrgspnsubrunlem1  33635